Adventures In PRIME NUMBER Land !!! ... (A Dummies Guide to Prime Numbers)

*** WARNING ***
This thread is of a mathematical nature and deals with the topic of PRIME NUMBERS.
If you're not interested in maths or are uncomfortable when dealing with maths oriented topics, then please don't bother reading further.
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Now, let me state from the start that I'm no mathematician by any stretch of the imagination but do have college level maths ability and ever since I
can remember (from a young age), I've always had an interest in prime numbers and how they seem to have a special place within the number system that
we use.

Primes on the surface appear to be just ordinary numbers but when looked at closely, they seem to take on a mysterious life of their own ... they give
the impression of being randomly dispersed amongst all the other numbers but then under closer inspection, seem to hint at having some kind of
pattern, some kind of underlying order that determines when and where they'll pop up amongst the other "ordinary" numbers.

This "pattern" or "order" has fascinated many, many people over the ages ... from Aristotle thru to Euler and others beyond. Some have been
mathematicians, others have been just amateur dabblers but all have shared the fascination and mysteriousness that's come to be associated with the
primes.

Besides being "intellectually and mathematically" intriguing, prime numbers actually play a very important role in modern cryptography and are used
to generate very secure digital keys that play a major role in such everyday activities as keeping electronic transactions between banks and stores
secure and safe from manipulation.

For those that aren't too clued up on what a prime number is, well that's easy to answer and no rocket science knowledge needed ... a prime number
is any number that can only be divided evenly by itself and the number 1.
In this case, when we refer to a "number", we're only talking about the positive integers e.g. 1, 2, 3, 4, 5, etc.
Now because EVERY number can be divided by 1, the definition of a prime number becomes even simpler ... a prime number is any number that can ONLY be
divided evenly by itself.

Examples of numbers that ARE prime:
7 ...... because it can be divided evenly ONLY by 7
13 .... because it can be divided evenly ONLY by 13
103 ... because it can be divided evenly ONLY by 103

Examples of numbers that are NOT prime:
8 ..... because it can be divided evenly by not only 8 but ALSO by 2 and 4
9 ..... because it can be divided evenly by not only 9 but ALSO by 3
45 ... because it can be divided evenly by not only 45 but ALSO by 3, 5, 9 and 15

So, let's take a look at the 1st 40 prime numbers, just to get a feel for them. The primes are coloured red .

but of course the list goes on and on as there's an infinite number of prime numbers.

You may have noticed that the number 1 wasn't included in the list above ... the reason being that mathematicians decided to exclude it was purely
because it could only be divided by 1, whereas all other primes could be divided by 1 AND themselves.
A small point but they get to make the rules up !

Now, looking at that list (above), you get the impression that there's no particular order to when and where the primes pop up, just randomly mixed
up with all the other numbers.
But, mathematicians believe that "where and when" the primes occur is in fact controlled by some kind of "rule" and not just thrown around
randomly. The search for this "rule" has occupied many people over the centuries and so far, not much progress has really been achieved in finding
this "rule".

So far, all the above is basic prime knowledge and can be obtained from any basic maths book.
What follows is, I hope, my own slant on the prime number system and what I've worked out thru my own dabblings over the years.
Perhaps it sheds some more light on how the primes work ... perhaps it doesn't.
If it does, then I'll be pleased to know that in my own way, I've contributed just a token amount of new incite !

Ok, the 1st thing I need to state and absolutely crucial to everything that subsequently follows in this thread, is that I believe the ENTIRE prime number system is BASED ON and COMPLETELY revolves around the number 24.

As they say, a picture is worth a thousand words.

To illustrate my statement above, lets create a number of concentric circles and divide each of them up into 24 segments. You can picture the 1st
circle as a clock with 24 hours ... starting from 1 and going around to 24.
The next outer circle can be pictured as a clock starting at 25 and going around for another 24 segments to finish at 48.
The third outer circle can be pictured as a clock starting at 49 and going around for another 24 segments to finish at 72.

So what we end up with is an infinite number of concentric circles, each divided up into 24 segments:
Circle 1 ---> 24 segments labeled 1 to 24
Circle 2 ---> 24 segments labeled 25 to 48
Circle 3 ---> 24 segments labeled 49 to 72
Circle 4 ---> 24 segments labeled 73 to 96
and so on for infinity.

So, in the above diagram, what I've essentially done is started counting numbers starting from 1 all the way to number 24, and put these numbers on
the 1st circle ... I've then continued counting for another 24 numbers (i.e. numbers 25 to 48) and put them on circle 2 ... I've then continued
counting for another 24 numbers (i.e. numbers 49 to 72) and put them on circle 3 ... and so on.
The 1st starting number of each subsequent circle (i.e. 25, then 49, then 73, etc ) I've lined up above number 1 (on the 1st circle).

Now, to help make the prime numbers stand out from all the other numbers, I've highlighted the primes in red.

Whew .... hope that makes things clearer for you

Spend a few moments examining the above diagram to become comfortable with what I've done so far before reading further.

Ready to continue ?

Because this thread is about the PRIME NUMBERS, lets make a modification to the above diagram and remove ALL numbers that are NOT prime ... giving a
less cluttered diagram.
Now, after removing every non-prime number, we end up with this amazing image.
EVERY prime number is to be found ONLY on one of the 8 blue coloured rays. So, even if you continued adding an infinite number of circles, you will
only find prime numbers appearing somewhere along extensions of these 8 rays !

A small amount of order has now been imposed on chaos !

DIAGRAM2:

Beautiful ... isn't it !!!

Now, lets amuse ourselves for a few minutes.

Let's say you wanted to know on which of the 8 rays a particular prime number will be found.

Really easy to do ... let's use a few examples.

Obviously, the 1st 8 primes (1, 5, 7, 11, 13, 17, 19 and 23) will be found on their correspondingly numbered rays e.g. 5 is on ray5; 13 is on ray13;
23 is on ray23).

What about prime number 31 ?
Divide 31 by 24 getting an answer. Use the answer value to the left of the decimal point and add 1 to obtain the circle number. Then multiply the
answer value to the right of the decimal point by 24 and you get the number of the ray that prime number 31 will be found on.

1. 31 / 24 = 1.2916666666666 (repeating)
2. For the circle number, use the number to the left of the decimal point and add 1
1 + 1 = 2
3. Now multiply the number to the right of the decimal point by 24 to get the ray number
0.2916666666666 x 24 = 7

Therefore prime number 31 will be found on circle 2, ray 7.

What about prime number 163 ?
1. 163 / 24 = 6.7916666666666 (repeating)
2. For the circle number, use the number to the left of the decimal point and add 1
6 + 1 = 7
3. Now multiply the number to the right of the decimal point by 24 to get the ray number
0.7916666666666 x 24 = 19

Therefore prime number 1,299,827 will be found on circle 54,160, ray 11.

(Yes, took a while but I have CONFIRMED the above result !)

By now you're probably asking "so what, what's the big deal working out the circle/ray that a prime number is on ?".

Well, because (until it's proven false) the above gives us a very fast and quick method for determining the "potential" primality of ANY given
number. In other words, pick ANY number you want, and two things can be immediately deduced ... either it's DEFINITELY NOT a prime number and
immediately eliminated ... or it's a "possible" prime number candidate.
If it's NOT a prime number, that's fine, otherwise other mathematical tests can be performed to CONFIRM whether its a DEFINITE prime number.
So, I guess it's a very quick method for eliminating numbers which are definitely non-prime.

Example:

We know number 99837 is NOT a prime but let's pretend we have no idea and let's
put it to the test.

1. 99837 / 24 = 4159.875
2. 4159 + 1 = 4160
3. 0.875 x 24 = 21

So number 99837 is located on ray 21 which is NOT one of the 8 rays that prime numbers are only to be found on.

Therefore, number 99837 is DEFINITELY NOT a prime number !

Go ahead and try any other numbers that you know are NOT primes and see if it continues to successfully eliminate them as non-prime.

Now, before we continue, there's something that needs to be cleared up.

In my opening post I mentioned that by convention, the number 1 was declared as NOT a prime even though like other members of the prime family, it can
only be divided by itself.
Also, number 2 is unique in all the infinite primes because it's the ONLY conventional prime that is an even number ... every other prime is odd.

So, according to "convention", the 1st two primes are the numbers 2 and 3.

But if you refer back to Diagram 2, you'll notice that firstly, the number 1 DOES appear on a prime number ray (ray1) and secondly, that neither the
number 2 or the number 3 appear on any of the 8 rays associated with the infinite amount of prime numbers.
We therefore seem to have a bit of a "conflict" here between what conventional prime number theory and my version have to say about these 1st three
prime numbers (i.e. numbers 1, 2 and 3).

Considering that I've managed to generate a geometrical model that successfully demonstrated a type of prime number distribution based on a very
simple rule that primes are generated around a base of 24, I'm prepared to go out on a limb and state that in my opinion (ONLY !!), that the number 1
should be reinstated as a prime and further that the numbers 2 and 3 should not be considered "true" primes as they do NOT fall on any of the rays
that the remaining infinite number of primes do fall on.
Also, we should be wary of considering the number 2 as prime because of it's unique status of being the ONLY conventional prime that is EVEN ... one
should rightly question why this single conventional prime should have such an "elevated and special" position compared to the infinitude of
primes.

Coincidentally, nature may even have "encoded" a clue pointing to this "base of 24" that all my work revolves around, in the three numbers located
between my 1st prime number (1) and my 2nd prime number (5). Between these two primes lie the numbers 2, 3 and 4.
The product of these three numbers (2 x 3 x 4) just happens to be 24.
Now, that certainly appeals to my sense of order !

By the way, the maths and my diagrams being based on 24 is known as Modular maths ... or sometimes referred to as "clock arithmetic" ... introduced
by the mathematician Carl Friedrich Gauss in 1801.
( en.wikipedia.org... )

In summary therefore, and as far as I can see, the initial five primes may actually be 1, 5, 7, 11, 13 and NOT the conventional 1, 2, 3, 5 and 7.

Ok, having got that little bit out of the way, let's take a look at other aspects of assuming that primes and their properties revolve around a base
of 24.

Take a look now at the following diagram.

DIAGRAM3.

The numbers you're seeing on ray1 are the squares of the 1st 5 primes.
Perhaps yet another indicator that numbers 2 and 3 should NOT be considered "true" primes as I've explained above.

In fact, no matter which prime you select, when you square it, the resulting value will ALWAYS appear on ray1.

Lets use the following prime numbers to illustrate this effect ... 41, 1009 and 10007.

Therefore, the square of the product of 601 and 1248007 is on Circle 23440763051065453, Ray1

Again, pick a few prime pairs yourself and confirm that the answer ALWAYS lies on Ray1.

Once we're aware of this property attached to squaring a prime and finding the answer ALWAYS on Ray1, we can generalize and now state that the
difference between ANY two primes that have been squared will ALWAYS be a multiple of 24.

What am I saying here ?

If you take ANY prime and square it, the answer will be somewhere on Ray1.
If you take ANY second prime and square it, the answer again will be somewhere on Ray1.
If you now subtract the smaller of the two answers from the larger of the two answers, the difference will ALWAYS be a multiple of the number 24.

But if you're still not convinced of the number 24 being at the very heart of the prime family, here's a final example of primes and their
dependency on the value 24.

Take ANY prime number and square it.
Then subtract 1 from the answer.
The new answer will be a multiple of 24.

In fact, in EVERY instance you'll find that again, the answer is ALWAYS a multiple of 24.

Example using prime number 7.

7 x 7 = 49 (squared)
49 - 1 = 48

48 = 24 x 2 = 24n (where n=2)

Example using prime number 82729.

82729 x 82729 = 6844087441
6844087441 - 1 = 6844087440

6844087440 = 24 x 285170310 = 24n (where n=285170310 )

Finally, convinced ???

Now we reach what I think is a very interesting part of the messing around I've been doing with primes.

In my opening post, I mentioned that primes are integral to the generation of highly secure keys used to encrypt all sorts of online communications
such as the transactions between banks and stores each time a credit card is used for a purchase or a transfer made online between two banks or an
automatic monthly direct debit is made from a bank account by a third party.

The basis of this encryption is quite straightforward ... two very large prime numbers are selected and multiplied together to create a resulting HUGE
number. This number is further processed to provide the final security "key" that will be used in the transaction.

The mathematical principal behind this is very simple to understand.

When you multiply any two numbers together, it's usually fairly trivial to take that final answer and work backwards to figure out the original two
numbers that were used. This process of obtaining the starting two numbers is call "factorization".

Example:
We can select the numbers 6 and 8, multiply them together to get an answer of 48.
Now using the 48, it's not difficult at all to work out that 6 and 8 (or even 4 and 12) were the original numbers used. Not very secure at all and I
certainly wouldn't trust any bank using such "weak" security !

But if instead two incredibly HUGE prime numbers were used that each consisted of hundreds of digits and multiplied them together, we'd get a monster
of an answer that no computer could possibly work out (factorize), in any kind of reasonable time frame, what the original 2 prime numbers were ... a
VERY secure system indeed !

So once you've multiplied those prime numbers together and got an answer, there really is no (currently) feasible way, if given the answer
alone, of working backwards and recovering the original two primes once more.

Which now leads me to introduce the following equation:

For the mathematically minded of you, this equation is just a variation of the general quartic formula of

This equation was derived from two simple observations, namely that
1. The square of any prime is ALWAYS on Ray1
2. The square of the products of any two primes is ALWAYS on Ray1.
3. That the difference between ANY two numbers on Ray1 will ALWAYS be a multiple of 24.

P1 represents the first prime number being used.
P2 represents the second prime number being used.
and
P1P2 represents the product obtained by multiplying the two prime numbers together.

As an example, lets assume that someone has selected two primes (unknown to us), multiplied them together and then provided us with the answer of
35.
Our task is to try to obtain the original two prime numbers that were used ... yes, I know this example is very trivial and could be done in your
head, but bear with me please).

Ok, the product of these two primes is 35.

From 2 above, taking the 35 and squaring it, we know that the answer of 1225 will be on Ray1.

From 1 above, we know that the squares of each of the two original primes must also be somewhere on Ray1.

And lastly, we know immediately that the circles that each of the squared primes are located on, must be less than the circle that 1225 is located on.
So when conducting our search for these two primes, we only need to search circles BELOW the circle that 1225 is on, thereby eliminating an enormous
amount of unnecessary search effort. Also, concentrating on Ray1 exclusively, means that again we have saved considerable search effort because we can
ignore the other 7 rays and the prime numbers on those rays.

So, using the given value of 35 and then squaring it to place it on Ray1, the original:

now looks like this:

For the moment, lets replace n with 1 (will explain why, shortly) which now gives us

At this point, we have no idea what prime number is represented by p1 in this equation.

But if we now proceed to graph the equation, we immediately see that there are 2 distinct roots ... one at the point (-5, 0) and one at (5, 0).
Ignoring the root at (-5, 0), we immediately see that point (5, 0) is in fact one of the prime numbers used and obviously the other prime number has
to be 7.
We have now determined that the original pair of prime numbers used were in fact 5 and 7 !

Lets try one more example ... this time the "unknown primes" will be 11 and 127 and when multiplied together, give us 1397.

So we're given the value 1397 and proceed to square it so it becomes associated with Ray1, and then we can insert the value of 1951609 into the
equation to give

As in the previous example, I'll now replace the variable n with the value 667 ... again, let me leave explaining the significance of n
until later.

At this point, we have no idea what prime number is represented by p1 in this equation.

But if we now proceed to graph the equation, we immediately see that there are 2 distinct roots ... one at the point (-11, 0) and one at (11, 0).
Ignoring the root at (-11, 0), we immediately see that point (11, 0) contains one of the prime numbers used and obviously the other prime number has
to be 127.
We have now determined that the original pair of prime numbers used were in fact 11 and 127 !

You'll recall that in the previous two examples, that I simply replaced the n variable with the value 1 (in the 1st example) and the value 672
(in the 2nd example).

So what was the significance of this n term ?

It actually tells us how many circles separate the square of the first prime and the square of the second prime.

In the 1st example using primes 5 and 7, the square of 5 (25) appears on circle2 and the square of 7 (49) appears on circle3.
Therefore circle3 is 1 circle higher than circle2 and therefore
(49 - 25) / 24 = 1 ... which is why I replaced n with the value 1.

In the 2nd example using primes 11 and 127, the square of 11 (121) appears on circle 6 and the square of 127 (16129) appears on circle673.
Therefore circle673 is 667 circles higher than circle6 and therefore
(16129 - 121) / 24 = 667 ... which is why I replaced n with the value 667.

Looks good, so far, don't you think ?
We seem to be able to determine the 2 unknown primes fairly easily if all we're given is their product after being multiplied together.

But did you notice the "spanner in the works" ?
For this to work, we need to KNOW right at the beginning, exactly how many circles apart the squares of each prime are on Ray1 ... in other words, we
need to know what "value" to replace the variable n with.
Unfortunately though, try as I can, I've been unable to come up with a working method to accomplish this.

Which is unfortunate, because if such a method can be devised, then we'll have a tool at our disposal that will facilitate the reverse extraction of
primes ... something that we're currently unable to do effectively or efficiently.

So, if any ATS members feel like becoming world famous (or infamous !) and would like to take a stab at solving this last "minor" hurdle, then
please feel free to give it a go.
After all, I'm not adverse to sharing the resultant publicity !

Oh, and by the way, all the info I've presented has already been intellectually copyrighted by myself :-)

Maths had never been one of my favourite subjects at school but I have to say that you've managed to convey the essence of prime numbers in a very
clear and understandable way ... at least I can say I've learned something new and quite interesting.
Also, it seems to me that you may actually be on to something here - keep working on it

However, sorry that my lack of knowledge won't be able to help you in any way

Not really, Calculus 2, Bus Calc, Statistics a few other courses in college. When I get a chance I will come back and absorb it. Mind is a little mush
yet, not enough coffee. Caffeine is good for the mind. So are warm hats-increase blood flow.

Actually, Euclid offered a proof of the infinity of primes around 2000 years ago ... amazing, isn't it !

Euclid stated that if you write down all known primes, you can always get a prime bigger than the last prime you used ... therefore, prime numbers are
infinite.

Proof:
Suppose 2,3,5,7,11,13,17 are the only primes you know.
You write a bigger number in this way by multiplying the primes you know together, than adding 1 to the answer.

N=2x3x5x7x11x13x17 + 1

This number is not a multiple of 2, 3, 5, 7, 11, 13 or 17 because performing division you would always have a remainder of one.
Therefore you have a couple of possibilities:

a. N is a NEW prime number itself
OR
b. N is a multiple of a prime greater than 17 (Your LAST highest prime).

In both cases you have the proof that a prime number exists that is larger than the largest prime you had before ... repeat this process as often as
you want and you'll ALWAYS find another prime larger than the largest prime you already had ... so primes are infinite.

WOW thats amazing! I have always loved math and everything I do when it comes to memorizing numbers I make up my own problem to remember it. I have
been teased for it but I have a great memory because of it! I am going to read this again

I also will show it to a friend who I know will enjoy it.
S&F for sure!

Originally posted by tauristercus
Sure, you could break the value 45361 down into the form 6(7560)+1, THEN take 7560 and divide by 24 to get 315 but I don't see how you would make the
24 connection directly from 6n+1 or 6n-1.

Does all this apply to only the base-10 number system or can the circles be transferred to octal, hexidecimal or other base number systems? Also,
consider base-24 and see if it works there too.

Prime numbers, to me, are not all that interesting. Is division of integers really meaningful outside of the world of math? We live in an analog
world (the irrational number Pi defines much of our spherical existence) - yet integers themselves are our counting method based on number of fingers
or "base-10". If we didn't devise a counting method - we wouldn't care as much about prime numbers.

Do primes correspond to anything such as the golden mean or golden ratio? Just wondering how prime numbers can help humanity better understand
ourselves.

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